This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334921 #12 Dec 21 2024 02:03:55
%S A334921 1,-2,2,0,-6,20,-48,96,-166,252,-340,416,-480,544,-544,0,2906,-13396,
%T A334921 44100,-121792,296860,-652808,1306560,-2377280,3879136,-5461952,
%U A334921 5892512,-2171520,-11699616,45871040,-114213888,228427776,-377994406,478195212,-252252460,-1013309824
%N A334921 Expansion of Phi(x) = (1/(1+x))*Product_{k>=0} (1-(x/(1+x))^2^k).
%C A334921 The Hankel transforms of Phi(x) and Phi(x^2) are identical. See Theorem 2.8 in Han paper.
%H A334921 Michel Marcus, <a href="/A334921/b334921.txt">Table of n, a(n) for n = 0..1000</a>
%H A334921 Guo-Niu Han, <a href="https://hal.archives-ouvertes.fr/hal-02125293">Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence</a>, Quaestiones Mathematicae, 2016, 39 (7), pp.895-909. hal-02125285.
%F A334921 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(-1)^(2*n-2*k+A000120(k)). See Theorem 2.8 in Han paper.
%F A334921 a(n) = 0 for positive terms of A024036.
%t A334921 a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * (-1)^(2*n - 2*k + DigitCount[k, 2, 1]), {k, 0, n}]; Array[a, 36, 0] (* _Amiram Eldar_, May 16 2020 *)
%o A334921 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(-1)^(2*n-2*k+hammingweight(k)));
%Y A334921 Cf. A000120 (Hamming weight of n), A024036 (4^n - 1).
%K A334921 sign
%O A334921 0,2
%A A334921 _Michel Marcus_, May 16 2020